Optimal. Leaf size=44 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right ) \]
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Rubi [A] time = 0.0101219, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {246, 245} \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^3\right )^m \, dx &=\left (\left (a+b x^3\right )^m \left (1+\frac{b x^3}{a}\right )^{-m}\right ) \int \left (1+\frac{b x^3}{a}\right )^m \, dx\\ &=x \left (a+b x^3\right )^m \left (1+\frac{b x^3}{a}\right )^{-m} \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )\\ \end{align*}
Mathematica [C] time = 0.155368, size = 196, normalized size = 4.45 \[ \frac{2^{-m} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-m} \left (\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}\right )^{-m} \left (a+b x^3\right )^m F_1\left (m+1;-m,-m;m+2;-\frac{i \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}{\sqrt{3} \sqrt [3]{a}},\frac{-\frac{2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt{3}+i}{3 i+\sqrt{3}}\right )}{\sqrt [3]{b} (m+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.25, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{3}+a \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{3} + a\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 15.3105, size = 34, normalized size = 0.77 \begin{align*} \frac{a^{m} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, - m \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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